(Generic region) Suppose in some region only $s_{0}$ and $t^{\lambda(v)}a_{v}s_{v}$ dominate, so the hypersurface locally looks like $\frac{s_{v}}{s_{0}}=\text{const}$. After normalising by powers of $t$ we may write this as $\{z_{0}=1\}$ in the coordinates $z_{0},z_{1},z_{2},z_{3}$ on the algebraic torus $(\mathbb{C}^{*})^{4}$. This model has $T^{3}$-symmetry under the diagonal action on $z_{1},z_{2},z_{3}$. The $T^{3}$-orbits are the natural candidate for approximate SYZ fibres. Thus we naturally look for a Kähler metric with potential $\phi=\phi(u_{1},u_{2},u_{3})$ depending only on the logarithms $u_{1}=\log|z_{1}|,u_{2}=\log|z_{2}|,u_{3}=\log|z_{3}|$. The holomorphic volume form $\Omega$ on the hypersurface is up to a scale factor given by

$$\frac{dz_{0}}{z_{0}}\wedge\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\frac{dz_{3}}{z_{3}}=d(z_{0}-1)\wedge\Omega,$$

namely $\Omega=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\frac{dz_{3}}{z_{3}}=d\log z_{1}\wedge d\log z_{2}\wedge d\log z_{3}$. The complex Monge-Ampère equation $(\sqrt{-1}\partial\bar{\partial}\phi)^{3}=\text{const}\sqrt{-1}\Omega\wedge\overline{\Omega}$ naturally reduces to the real Monge-Ampère equation $\det(\frac{\partial^{2}\phi}{\partial u_{i}\partial u_{j}})=\text{const}$. One can further calculate that such regions take up most of the volume measure on $X_{t}$, thus lending some evidence for the SYZ conjecture. We remark that the description only applies to local regions so the metrics are not complete.

The real Monge-Ampère equation governs also the region near the intersection of $X_{t}$ with a smooth component of the toric boundary. Suppose after normalising by powers of $t$, the dominant monomials are $z_{0}^{-1},z_{0}^{-1}z_{1},z_{0}^{-1}z_{2},z_{0}^{-1}z_{3},1$ in the coordinates $z_{0},z_{1},z_{2},z_{3}$ on the algebraic torus $(\mathbb{C}^{*})^{4}$, so the hypersurface has the local complex geometric model $\{z_{0}^{-1}(1+z_{1}+z_{2}+z_{3})+1=0\}\subset(\mathbb{C}^{*})^{4}$, or equivalently $-z_{0}=1+z_{1}+z_{2}+z_{3}$. The adjunction formula

$$\frac{dz_{0}}{z_{0}}\wedge\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\frac{dz_{3}}{z_{3}}=d(z_{0}^{-1}(1+z_{1}+z_{2}+z_{3})+1)\wedge\Omega$$

leads to $\Omega=\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\frac{dz_{3}}{z_{3}}$ as before. The diagonal $T^{3}$-action on $z_{1},z_{2},z_{3}$ provides the candidate for an approximate SYZ fibration, and a solution to the real Monge-Ampère equation in the $\log|z_{1}|,\log|z_{2}|,\log|z_{3}|$ coordinates induces a local Calabi-Yau metric.

Suppose after normalising by powers of $t$, the dominant monomials are $(z_{1}z_{2})^{-1},-(z_{1}z_{2})^{-1}z_{3},-1$, so the hypersurface admits the local complex geometric model $\{(z_{1}z_{2})^{-1}(1-z_{3})=1\}\subset(\mathbb{C}^{*})^{4}_{z_{0},z_{1},z_{2},z_{3}}$, or equivalently $z_{1}z_{2}=1-z_{3}$. This happens near the intersection of two smooth components of the toric boundary. Up to numerical factors, the holomorphic volume form is given by

$$\frac{dz_{0}}{z_{0}}\wedge\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge\frac{dz_{3}}{z_{3}}=d((z_{1}z_{2})^{-1}(1-z_{3})-1)\wedge\Omega,$$

namely $\Omega=d\log z_{0}\wedge\frac{dz_{1}\wedge dz_{2}}{z_{3}}$. This model has a natural $T^{2}$-symmetry: one $S^{1}$ acts trivially on $z_{1},z_{2},z_{3}$ and rotates $z_{0}$, while the other $S^{1}$ acts trivially on $z_{0},z_{3}$ and diagonally on $z_{1},z_{2}$. The model is intimately related to the Ooguri-Vafa metric (cf. Section 1.3.2), and we expect this region to coincide with the neighbourhood of edges in the Gross-Ruan-Joyce picture.

The positive vertex $M^{+}$ describes a neighbourhood of the point $(0,0,0,1)$ inside $\{z_{0}z_{1}z_{2}=1-z_{3}\}\subset\mathbb{C}^{3}\times\mathbb{C}^{*}_{z_{3}}$. In the toric hypersurface picture, we are looking at a region where the dominating monomials are up to scale factors $(z_{0}z_{1}z_{2})^{-1},-z_{3}(z_{0}z_{1}z_{2})^{-1},1$, so the defining equation of $X_{t}$ is approximately $(z_{0}z_{1}z_{2})^{-1}(1-z_{3})=1$ once we absorb the scale factors into $z_{i}$. The holomorphic volume form $\Omega$ is up to constant given by

$$-\frac{\sqrt{-1}}{2\pi}d\log z_{0}\wedge d\log z_{1}\wedge d\log z_{2}\wedge d\log z_{3}=d((z_{0}z_{1}z_{2})^{-1}(1-z_{3})-1)\wedge\Omega,$$

or equivalently $\Omega=-\frac{\sqrt{-1}}{2\pi}\frac{1}{z_{3}}dz_{0}\wedge dz_{1}\wedge dz_{2}$. An important feature of this model is the diagonal $T^{2}$-symmetry:

$$e^{i\theta_{1}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{1}}z_{0},e^{i\theta_{1}}z_{1},z_{2}),\quad e^{i\theta_{2}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{2}}z_{0},z_{1},e^{i\theta_{2}}z_{2}).$$

We have $\Omega(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=-\frac{\sqrt{-1}}{2\pi}d\log z_{3}=d\eta$, where $\eta=-\frac{\sqrt{-1}}{2\pi}\log(z_{3})$ is a holomorphic coordinate with period 1, and takes the value zero at $z_{0}=z_{1}=z_{2}=0$. The relation between this complex geometric perspective and the topological picture in Section 1.1.3 is perhaps clearest with the generalised Gibbons-Hawking construction in mind (cf. Section 1.2 below). Essentially $M^{+}$ is a singular $T^{2}$-bundle over a 4-dimensional base contained in $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$, where $\mu_{1},\mu_{2}$ are the $T^{2}$-moment maps normalised to have value 0 at $z_{0}=z_{1}=z_{2}=0$. We shall notice that the discriminant locus $\mathfrak{D}\times\{0\}$ of this singular $T^{2}$-bundle is not sensitive to the choice of the Kähler form (cf. Lemma 1.6 and its ensuing Remark). The normalising constant on $\Omega$ imply that the SYZ $T^{3}$-fibres have $\int_{T^{3}}\Omega=4\pi^{2}$.

The negative vertex $M^{-}$ describes an open subset inside $\{z_{3}z_{4}=1-z_{1}-z_{2}\}\subset\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{C}^{2}_{z_{3},z_{4}}$. In the toric hypersurface picture, we are looking at a region where the dominating monomials are up to scale factors just $(z_{3}z_{4})^{-1}$, $z_{1}(z_{3}z_{4})^{-1}$, $z_{2}(z_{3}z_{4})^{-1}$ and 1, so the defining equation of $X_{t}$ is approximately $(1-z_{1}-z_{2})(z_{3}z_{4})^{-1}-1=0$ once we absorb the scale factors into $z_{i}$. The holomorphic volume form $\Omega$ is given up to constant by

$$\frac{\sqrt{-1}}{4\pi^{2}}d\log z_{1}\wedge d\log z_{2}\wedge d\log z_{3}\wedge d\log z_{4}=d\left((1-z_{1}-z_{2})(z_{3}z_{4})^{-1}-1\right)\wedge\Omega,$$

or equivalently $\Omega=\frac{-\sqrt{-1}}{4\pi^{2}}\frac{1}{z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}.$ This model has $S^{1}$-symmetry:

$$e^{i\theta}\cdot(z_{1},z_{2},z_{3},z_{4})=(z_{1},z_{2},e^{i\theta}z_{3},e^{-i\theta}z_{4}).$$

Hence $M^{-}$ is a singular $S^{1}$-bundle over $\mathbb{R}_{\mu}\times\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}$, where $\mu$ is the $S^{1}$-moment coordinate which takes the value zero on the singular locus $\{z_{3}=z_{4}=0\}$ (notice that the degeneracy of the $S^{1}$ factor implies that the moment map is constant on this singular locus for any choice of Kähler form). This agrees with the modified topological description in Section 1.1.5. We calculate

$$\iota_{\frac{\partial}{\partial\theta}}\Omega=-\frac{1}{4\pi^{2}}d\log z_{1}\wedge d\log z_{2}=d\eta_{1}\wedge d\eta_{2},$$

where the logarithmic coordinates $\eta_{p}=\frac{1}{2\pi\sqrt{-1}}\log z_{p}$ for $p=1,2$ have period 1. The $\arg z_{1},\arg z_{2}$ coordinates provide a family of 2-tori in $\mathbb{R}\times\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}$, and the restriction of the $S^{1}$-bundle over these 2-tori defines a family of 3-tori. The normalising constant on $\Omega$ imply that $\int_{T^{3}}\Omega=2\pi$.